Distributed Learning for Stochastic Generalized Nash Equilibrium Problems

TL;DR

This paper develops distributed stochastic gradient algorithms for generalized Nash equilibrium problems with coupled constraints, demonstrating convergence to equilibrium in a networked environment with randomness.

Contribution

It introduces three novel stochastic gradient strategies with heterogeneous step-sizes for solving GNEPs in a distributed, online setting with theoretical convergence guarantees.

Findings

Algorithms converge within $O( ext{step-size})$ of the Nash equilibrium.

Heterogeneous step-sizes are effective in the distributed setting.

Application to network Cournot competition illustrates practical utility.

Abstract

This work examines a stochastic formulation of the generalized Nash equilibrium problem (GNEP) where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully-distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within $O(\mu_\text{max})$, for small step-size value $\mu_\text{max}$ and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem.